Automatically Find and Download file from link in Email

I have an email account that receives contact forms from my website (that my clients send me). The contact forms always have a file submitted with them. I'm trying to automatically download the file from each contact form email that comes in. Is this possible? If so, how?

I'm not sure if I should be dealing with Mail, deal with the Gmail account in Chrome, or if there's some third-party app that handles this well like Zapier, for example.

I see the process as:

  • receive an email
  • search email and find the URL to the file in the email (probably named differently each time, but maybe I can impose a naming convention?)
  • open it (optional?)
  • download the file

Thanks for your help!